# K-theory of free G-sets and the classifying space, and generalization [reference-request]

Let $$G$$ be a finite group, $$\mathcal{G}^0$$ be the category of finite free $$G$$-sets and isomorphisms between them. Then $$\mathcal{G}^0$$ is a symmetric monoidal category with respect to the disjoint union, so we can talk about its $$K$$-theory, which is homotopy equivalent to $$\Sigma ^{\infty}BG_+$$.

My first question is, where in the literature can I find this, preferably explicitly stated in this way?

Now, consider the category $$\mathcal{G}$$ who has unique object and morphisms are elements of $$G$$. Then we can identify $$G$$-sets with functors from $$\mathcal{G}$$ to $$Sets$$. Thus the statement above can be reformulated as follows.

$$\Sigma ^{\infty}|Nerve(\mathcal{G})|_+\simeq K(F^0(\mathcal{G},Sets))$$ where $$F^0(A,B)$$ denote the category whose objects are "free" functor from $$\mathcal{G}$$ to $$Sets$$. And by free, we mean the objects that are in the essential image of the left adjoint of the "forgetful" functor to $$Sets$$.

Now my second question is: is there any known sufficient condition on the category $$\mathcal{C}$$ and its object $$C$$ so that we have $$\Sigma ^{\infty}|Nerve(\mathcal{C})|_+\simeq K(F^0(\mathcal{C},Sets)),$$ where the notation is just as in above except we use the evaluation at $$C$$ instead of the forgetful functor?

• I thought $K$-theory of a category is the one of its classifying space. But, it seems you are identifying the suspension spectrum of the classifying space as the $K$-theory of your category?! – user51223 Jan 16 at 20:23
• c.f. my comment on the answer by John Klein. – user43326 Jan 17 at 6:56

The general point is just that if $$\mathcal{U}$$ is equivalent to the free symmetric monoidal category $$F\mathcal{C}$$ generated by $$\mathcal{C}$$ then $$K(\mathcal{U})\simeq\Sigma^\infty_+\mathcal{C}$$. Here $$F\mathcal{C}$$ can be constructed as the category of pairs $$(X,C)$$, where $$X$$ is a finite set and $$C\in\prod_{x\in X}\text{obj}(\mathcal{C})$$. A morphism from $$(X,C)$$ to $$(Y,D)$$ consists of a bijection $$\sigma\colon X\to Y$$ together with a family of $$\mathcal{C}$$-morphisms $$C_x\to D_{\sigma(x)}$$ for all $$x\in X$$. It's not hard to identify the groupoid $$\mathcal{F}G$$ of finite $$G$$-sets with the free symmetric monoidal category generated by subgroupoid $$\text{Orb}(G)$$ of transitive $$G$$-sets. We can choose subgroups $$H_1,\dotsc,H_m$$ containing one representative of each conjugacy class, and let $$W_k=N_G(H_k)/H_k$$ denote the Weyl group, considered as a one-object groupoid. Then $$\text{Orb}(G)$$ is equivalent to the coproduct of the groupoids $$W_k$$, so we get $$K(\mathcal{F}G) \simeq \Sigma^\infty_+ B\text{Orb}(G) \simeq \bigvee_k \Sigma^\infty_+ BW_k$$ This is the tom Dieck splitting. If we restrict attention to the subcategory $$\mathcal{F}_1G$$ of finite free $$G$$-sets, we find that this is freely generated by the free orbit $$G/1$$, whose $$G$$-equivariant automorphism group is the Weyl group $$W1=G$$, so we get $$K(\mathcal{F}_1G)=\Sigma^\infty_+BG$$. There are quite a few interesting results in this vein, but unfortunately I do not think that there is a good source in the literature.

• Maybe I am asking something dumb, but how does "The general point" work? – user43326 Jan 17 at 7:37

I believe you meant to write $$Q(BG_+)$$ in the first paragraph of your post, where $$Q = \Omega^\infty\Sigma^\infty$$. The this result is really a folk theorem and is sometimes called the "Barratt-Priddy-Quillen-Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the "group-completion'' theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $$G$$-sets and their isomorphisms defines a topological monoid $$M$$. The group completion theorem tells us in this case that $$\Omega B M$$ coincides with $$Q(BG_+)$$.

• Actually I really meant $BG_+$ because some author use the word $k$-theory to mean the associated spectrum (infinite delooping) and not the infinite loop space. – user43326 Jan 17 at 6:48
• @user43326 the K-theory of finite free $G$-sets is not $BG$, it is $Q(BG_+)$. Here is a simple reason why it can't be $BG$: the latter is not in general an infiinite loop space (but the $K$-theory is an infinite loop space). By the way, $G$ does not need to be finite in the Barratt-Priddy-Quillen-Segal theorem. – John Klein Jan 17 at 12:14
• that is exactly what is I am saying, for some authors it is the spectrum and not the infinite loop space... – user43326 Jan 17 at 15:15
• Oh, thank you, I had missed that. In most of my writing, I identify a based space with its suspension spectrum... Corrected now. – user43326 Jan 19 at 8:08
• @JohnKlein different people use different notation. I would quite often leave the $\Sigma^\infty$ implicit in this kind of context. – Neil Strickland Jan 19 at 9:18